Symplectic Invariants And Hamiltonian Dynamics

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Symplectic Invariants and Hamiltonian Dynamics

Author : Helmut Hofer
Publisher : Birkhäuser
ISBN 13 : 3034885407
Total Pages : 356 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Symplectic Invariants and Hamiltonian Dynamics by : Helmut Hofer

Download or read book Symplectic Invariants and Hamiltonian Dynamics written by Helmut Hofer and published by Birkhäuser. This book was released on 2012-12-06 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: Analysis of an old variational principal in classical mechanics has established global periodic phenomena in Hamiltonian systems. One of the links is a class of sympletic invariants, called sympletic capacities, and these invariants are the main theme of this book. Topics covered include basic sympletic geometry, sympletic capacities and rigidity, sympletic fixed point theory, and a survey on Floer homology and sympletic homology.

Lectures on Dynamical Systems

Author : Eduard Zehnder
Publisher : European Mathematical Society
ISBN 13 : 9783037190814
Total Pages : 372 pages
Book Rating : 4.1/5 (98 download)

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Book Synopsis Lectures on Dynamical Systems by : Eduard Zehnder

Download or read book Lectures on Dynamical Systems written by Eduard Zehnder and published by European Mathematical Society. This book was released on 2010 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book originated from an introductory lecture course on dynamical systems given by the author for advanced students in mathematics and physics at ETH Zurich. The first part centers around unstable and chaotic phenomena caused by the occurrence of homoclinic points. The existence of homoclinic points complicates the orbit structure considerably and gives rise to invariant hyperbolic sets nearby. The orbit structure in such sets is analyzed by means of the shadowing lemma, whose proof is based on the contraction principle. This lemma is also used to prove S. Smale's theorem about the embedding of Bernoulli systems near homoclinic orbits. The chaotic behavior is illustrated in the simple mechanical model of a periodically perturbed mathematical pendulum. The second part of the book is devoted to Hamiltonian systems. The Hamiltonian formalism is developed in the elegant language of the exterior calculus. The theorem of V. Arnold and R. Jost shows that the solutions of Hamiltonian systems which possess sufficiently many integrals of motion can be written down explicitly and for all times. The existence proofs of global periodic orbits of Hamiltonian systems on symplectic manifolds are based on a variational principle for the old action functional of classical mechanics. The necessary tools from variational calculus are developed. There is an intimate relation between the periodic orbits of Hamiltonian systems and a class of symplectic invariants called symplectic capacities. From these symplectic invariants one derives surprising symplectic rigidity phenomena. This allows a first glimpse of the fast developing new field of symplectic topology.

Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces

Author : Victor Guillemin
Publisher : Springer Science & Business Media
ISBN 13 : 1461202698
Total Pages : 158 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces by : Victor Guillemin

Download or read book Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces written by Victor Guillemin and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: The action of a compact Lie group, G, on a compact sympletic manifold gives rise to some remarkable combinatorial invariants. The simplest and most interesting of these is the moment polytopes, a convex polyhedron which sits inside the dual of the Lie algebra of G. One of the main goals of this monograph is to describe what kinds of geometric information are encoded in this polytope. This book is addressed to researchers and can be used as a semester text.

Symplectic Topology and Floer Homology: Volume 2, Floer Homology and its Applications

Author : Yong-Geun Oh
Publisher : Cambridge University Press
ISBN 13 : 1316381390
Total Pages : 471 pages
Book Rating : 4.3/5 (163 download)

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Book Synopsis Symplectic Topology and Floer Homology: Volume 2, Floer Homology and its Applications by : Yong-Geun Oh

Download or read book Symplectic Topology and Floer Homology: Volume 2, Floer Homology and its Applications written by Yong-Geun Oh and published by Cambridge University Press. This book was released on 2015-08-27 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: Published in two volumes, this is the first book to provide a thorough and systematic explanation of symplectic topology, and the analytical details and techniques used in applying the machinery arising from Floer theory as a whole. Volume 2 provides a comprehensive introduction to both Hamiltonian Floer theory and Lagrangian Floer theory, including many examples of their applications to various problems in symplectic topology. The first volume covered the basic materials of Hamiltonian dynamics and symplectic geometry and the analytic foundations of Gromov's pseudoholomorphic curve theory. Symplectic Topology and Floer Homology is a comprehensive resource suitable for experts and newcomers alike.

The Breadth of Symplectic and Poisson Geometry

Author : Jerrold E. Marsden
Publisher : Springer Science & Business Media
ISBN 13 : 0817644199
Total Pages : 666 pages
Book Rating : 4.8/5 (176 download)

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Book Synopsis The Breadth of Symplectic and Poisson Geometry by : Jerrold E. Marsden

Download or read book The Breadth of Symplectic and Poisson Geometry written by Jerrold E. Marsden and published by Springer Science & Business Media. This book was released on 2007-07-03 with total page 666 pages. Available in PDF, EPUB and Kindle. Book excerpt: * The invited papers in this volume are written in honor of Alan Weinstein, one of the world’s foremost geometers * Contributions cover a broad range of topics in symplectic and differential geometry, Lie theory, mechanics, and related fields * Intended for graduate students and working mathematicians, this text is a distillation of prominent research and an indication of future trends in geometry, mechanics, and mathematical physics

Lectures on Symplectic Geometry

Author : Ana Cannas da Silva
Publisher : Springer
ISBN 13 : 354045330X
Total Pages : 220 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Lectures on Symplectic Geometry by : Ana Cannas da Silva

Download or read book Lectures on Symplectic Geometry written by Ana Cannas da Silva and published by Springer. This book was released on 2004-10-27 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.

Function Theory on Symplectic Manifolds

Author : Leonid Polterovich
Publisher : American Mathematical Soc.
ISBN 13 : 147041693X
Total Pages : 282 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Function Theory on Symplectic Manifolds by : Leonid Polterovich

Download or read book Function Theory on Symplectic Manifolds written by Leonid Polterovich and published by American Mathematical Soc.. This book was released on 2014 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a book on symplectic topology, a rapidly developing field of mathematics which originated as a geometric tool for problems of classical mechanics. Since the 1980s, powerful methods such as Gromov's pseudo-holomorphic curves and Morse-Floer theory on loop spaces gave rise to the discovery of unexpected symplectic phenomena. The present book focuses on function spaces associated with a symplectic manifold. A number of recent advances show that these spaces exhibit intriguing properties and structures, giving rise to an alternative intuition and new tools in symplectic topology. The book provides an essentially self-contained introduction into these developments along with applications to symplectic topology, algebra and geometry of symplectomorphism groups, Hamiltonian dynamics and quantum mechanics. It will appeal to researchers and students from the graduate level onwards.

Generic Hamiltonian Dynamical Systems are Neither Integrable nor Ergodic

Author : Lawrence Markus
Publisher : American Mathematical Soc.
ISBN 13 : 0821818449
Total Pages : 58 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Generic Hamiltonian Dynamical Systems are Neither Integrable nor Ergodic by : Lawrence Markus

Download or read book Generic Hamiltonian Dynamical Systems are Neither Integrable nor Ergodic written by Lawrence Markus and published by American Mathematical Soc.. This book was released on 1974 with total page 58 pages. Available in PDF, EPUB and Kindle. Book excerpt: This memoir gives an introduction to Hamiltonian dynamical systems on symplectic manifolds, including definitions of Hamiltonian vector fields, Poisson brackets, integrals of motion, complete integrability, and ergodicity. A particularly complete treatment of action-angle coordinates is given. Historical background into the question of ergodicity and integrability in Hamiltonian systems is also given.

Hamiltonian Dynamics

Author : Gaetano Vilasi
Publisher : World Scientific
ISBN 13 : 9810233086
Total Pages : 457 pages
Book Rating : 4.8/5 (12 download)

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Book Synopsis Hamiltonian Dynamics by : Gaetano Vilasi

Download or read book Hamiltonian Dynamics written by Gaetano Vilasi and published by World Scientific. This book was released on 2001 with total page 457 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is both a textbook and a monograph. It is partially based on a two-semester course, held by the author for third-year students in physics and mathematics at the University of Salerno, on analytical mechanics, differential geometry, symplectic manifolds and integrable systems.As a textbook, it provides a systematic and self-consistent formulation of Hamiltonian dynamics both in a rigorous coordinate language and in the modern language of differential geometry. It also presents powerful mathematical methods of theoretical physics, especially in gauge theories and general relativity.As a monograph, the book deals with the advanced research topic of completely integrable dynamics, with both finitely and infinitely many degrees of freedom, including geometrical structures of solitonic wave equations.

Symplectic Topology and Measure Preserving Dynamical Systems

Author : Albert Fathi
Publisher : American Mathematical Soc.
ISBN 13 : 0821848925
Total Pages : 192 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Symplectic Topology and Measure Preserving Dynamical Systems by : Albert Fathi

Download or read book Symplectic Topology and Measure Preserving Dynamical Systems written by Albert Fathi and published by American Mathematical Soc.. This book was released on 2010-04-09 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: The papers in this volume were presented at the AMS-IMS-SIAM Joint Summer Research Conference on Symplectic Topology and Measure Preserving Dynamical Systems held in Snowbird, Utah in July 2007. The aim of the conference was to bring together specialists of symplectic topology and of measure preserving dynamics to try to connect these two subjects. One of the motivating conjectures at the interface of these two fields is the question of whether the group of area preserving homeomorphisms of the 2-disc is or is not simple. For diffeomorphisms it was known that the kernel of the Calabi invariant is a normal proper subgroup, so the group of area preserving diffeomorphisms is not simple. Most articles are related to understanding these and related questions in the framework of modern symplectic topology.

Momentum Maps and Hamiltonian Reduction

Author : Juan-Pablo Ortega
Publisher : Springer Science & Business Media
ISBN 13 : 9780817643072
Total Pages : 544 pages
Book Rating : 4.6/5 (43 download)

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Book Synopsis Momentum Maps and Hamiltonian Reduction by : Juan-Pablo Ortega

Download or read book Momentum Maps and Hamiltonian Reduction written by Juan-Pablo Ortega and published by Springer Science & Business Media. This book was released on 2003-12-16 with total page 544 pages. Available in PDF, EPUB and Kindle. Book excerpt: * Winner of the Ferran Sunyer i Balaguer Prize in 2000. * Reviews the necessary prerequisites, beginning with an introduction to Lie symmetries on Poisson and symplectic manifolds. * Currently in classroom use in Europe. * Can serve as a resource for graduate courses and seminars in Hamiltonian mechanics and symmetry, symplectic and Poisson geometry, Lie theory, mathematical physics, and as a comprehensive reference resource for researchers.

Notes on Dynamical Systems

Author : Jurgen Moser
Publisher : American Mathematical Soc.
ISBN 13 : 0821835777
Total Pages : 266 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Notes on Dynamical Systems by : Jurgen Moser

Download or read book Notes on Dynamical Systems written by Jurgen Moser and published by American Mathematical Soc.. This book was released on 2005 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to the field of dynamical systems, in particular, to the special class of Hamiltonian systems. The authors aimed at keeping the requirements of mathematical techniques minimal but giving detailed proofs and many examples and illustrations from physics and celestial mechanics. After all, the celestial $N$-body problem is the origin of dynamical systems and gave rise in the past to many mathematical developments. Jurgen Moser (1928-1999) was a professor atthe Courant Institute, New York, and then at ETH Zurich. He served as president of the International Mathematical Union and received many honors and prizes, among them the Wolf Prize in mathematics. Jurgen Moser is the author of several books, among them Stable and Random Motions in DynamicalSystems. Eduard Zehnder is a professor at ETH Zurich. He is coauthor with Helmut Hofer of the book Symplectic Invariants and Hamiltonian Dynamics. Information for our distributors: Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.

Hamiltonian Dynamics

Author : Gaetano Vilasi
Publisher : World Scientific
ISBN 13 : 9814496731
Total Pages : 457 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Hamiltonian Dynamics by : Gaetano Vilasi

Download or read book Hamiltonian Dynamics written by Gaetano Vilasi and published by World Scientific. This book was released on 2001-03-09 with total page 457 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is both a textbook and a monograph. It is partially based on a two-semester course, held by the author for third-year students in physics and mathematics at the University of Salerno, on analytical mechanics, differential geometry, symplectic manifolds and integrable systems.As a textbook, it provides a systematic and self-consistent formulation of Hamiltonian dynamics both in a rigorous coordinate language and in the modern language of differential geometry. It also presents powerful mathematical methods of theoretical physics, especially in gauge theories and general relativity.As a monograph, the book deals with the advanced research topic of completely integrable dynamics, with both finitely and infinitely many degrees of freedom, including geometrical structures of solitonic wave equations.

Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory

Author : Kenji Fukaya
Publisher : American Mathematical Soc.
ISBN 13 : 1470436256
Total Pages : 266 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory by : Kenji Fukaya

Download or read book Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory written by Kenji Fukaya and published by American Mathematical Soc.. This book was released on 2019-09-05 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper the authors first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entov-Polterovich theory of spectral symplectic quasi-states and quasi-morphisms by incorporating bulk deformations, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher in a slightly less general context. Then the authors explore various applications of these enhancements to the symplectic topology, especially new construction of symplectic quasi-states, quasi-morphisms and new Lagrangian intersection results on toric and non-toric manifolds. The most novel part of this paper is its use of open-closed Gromov-Witten-Floer theory and its variant involving closed orbits of periodic Hamiltonian system to connect spectral invariants (with bulk deformation), symplectic quasi-states, quasi-morphism to the Lagrangian Floer theory (with bulk deformation). The authors use this open-closed Gromov-Witten-Floer theory to produce new examples. Using the calculation of Lagrangian Floer cohomology with bulk, they produce examples of compact symplectic manifolds which admits uncountably many independent quasi-morphisms . They also obtain a new intersection result for the Lagrangian submanifold in .

Symplectic Geometry and Topology

Author : Yakov Eliashberg
Publisher : American Mathematical Soc.
ISBN 13 : 9780821886892
Total Pages : 452 pages
Book Rating : 4.8/5 (868 download)

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Book Synopsis Symplectic Geometry and Topology by : Yakov Eliashberg

Download or read book Symplectic Geometry and Topology written by Yakov Eliashberg and published by American Mathematical Soc.. This book was released on 2004 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: Symplectic geometry has its origins as a geometric language for classical mechanics. But it has recently exploded into an independent field interconnected with many other areas of mathematics and physics. The goal of the IAS/Park City Mathematics Institute Graduate Summer School on Symplectic Geometry and Topology was to give an intensive introduction to these exciting areas of current research. Included in this proceedings are lecture notes from the following courses: Introductionto Symplectic Topology by D. McDuff; Holomorphic Curves and Dynamics in Dimension Three by H. Hofer; An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by C. Taubes; Lectures on Floer Homology by D. Salamon; A Tutorial on Quantum Cohomology by A. Givental; Euler Characteristicsand Lagrangian Intersections by R. MacPherson; Hamiltonian Group Actions and Symplectic Reduction by L. Jeffrey; and Mechanics: Symmetry and Dynamics by J. Marsden. Information for our distributors: Titles in this series are copublished with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

The Floer Memorial Volume

Author : Helmut Hofer
Publisher : Birkhäuser
ISBN 13 : 3034892179
Total Pages : 688 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis The Floer Memorial Volume by : Helmut Hofer

Download or read book The Floer Memorial Volume written by Helmut Hofer and published by Birkhäuser. This book was released on 2012-12-06 with total page 688 pages. Available in PDF, EPUB and Kindle. Book excerpt: Andreas Floer died on May 15, 1991 an untimely and tragic death. His visions and far-reaching contributions have significantly influenced the developments of mathematics. His main interests centered on the fields of dynamical systems, symplectic geometry, Yang-Mills theory and low dimensional topology. Motivated by the global existence problem of periodic solutions for Hamiltonian systems and starting from ideas of Conley, Gromov and Witten, he developed his Floer homology, providing new, powerful methods which can be applied to problems inaccessible only a few years ago. This volume opens with a short biography and three hitherto unpublished papers of Andreas Floer. It then presents a collection of invited contributions, and survey articles as well as research papers on his fields of interest, bearing testimony of the high esteem and appreciation this brilliant mathematician enjoyed among his colleagues. Authors include: A. Floer, V.I. Arnold, M. Atiyah, M. Audin, D.M. Austin, S.M. Bates, P.J. Braam, M. Chaperon, R.L. Cohen, G. Dell' Antonio, S.K. Donaldson, B. D'Onofrio, I. Ekeland, Y. Eliashberg, K.D. Ernst, R. Finthushel, A.B. Givental, H. Hofer, J.D.S. Jones, I. McAllister, D. McDuff, Y.-G. Oh, L. Polterovich, D.A. Salamon, G.B. Segal, R. Stern, C.H. Taubes, C. Viterbo, A. Weinstein, E. Witten, E. Zehnder.

Lagrangian and Hamiltonian Dynamics

Author : Peter Mann
Publisher : Oxford University Press
ISBN 13 : 0192555413
Total Pages : 544 pages
Book Rating : 4.1/5 (925 download)

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Book Synopsis Lagrangian and Hamiltonian Dynamics by : Peter Mann

Download or read book Lagrangian and Hamiltonian Dynamics written by Peter Mann and published by Oxford University Press. This book was released on 2018-05-10 with total page 544 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introductory textbook exploring the subject of Lagrangian and Hamiltonian dynamics, with a relaxed and self-contained setting. Lagrangian and Hamiltonian dynamics is the continuation of Newton's classical physics into new formalisms, each highlighting novel aspects of mechanics that gradually build in complexity to form the basis for almost all of theoretical physics. Lagrangian and Hamiltonian dynamics also acts as a gateway to more abstract concepts routed in differential geometry and field theories and can be used to introduce these subject areas to newcomers. Journeying in a self-contained manner from the very basics, through the fundamentals and onwards to the cutting edge of the subject, along the way the reader is supported by all the necessary background mathematics, fully worked examples, thoughtful and vibrant illustrations as well as an informal narrative and numerous fresh, modern and inter-disciplinary applications. The book contains some unusual topics for a classical mechanics textbook. Most notable examples include the 'classical wavefunction', Koopman-von Neumann theory, classical density functional theories, the 'vakonomic' variational principle for non-holonomic constraints, the Gibbs-Appell equations, classical path integrals, Nambu brackets and the full framing of mechanics in the language of differential geometry.